8.2 A parallel plate capacitor (Fig. 8.7) made of circular plates each of radius R = 6.0 cm has a capacitance C = 100 pF. The capacitor is connected to a 230 V ac supply with a (angular) frequency of 300 rad s–1.
(a) What is the rms value of the conduction current?
(b) Is the conduction current equal to the displacement current?
(c) Determine the amplitude of B at a point 3.0 cm from the axis between the plates.
8.2 Radius of each circular plate, R = 6.0 cm = 0.06 m
Capacitance of parallel capacitor, C = 100 pF = 100 F
Supply voltage, V = 230 V
Angular frequency, = 300 rad/s
rms value of the conduction current, I = , where
Hence, I = V = 230 100 = 6.9 A = 6.9
Yes, the conduction current is equal to the displacement current.
Magnetic field is given as, B = , where
= Free space permeability = 4 N
= Maximum value of current =
r = distance between two plates on the axis = 3.0 cm = 0.03 m
then, B = 6.9 = 1.626&
...more
8.2 Radius of each circular plate, R = 6.0 cm = 0.06 m
Capacitance of parallel capacitor, C = 100 pF = 100 F
Supply voltage, V = 230 V
Angular frequency, = 300 rad/s
rms value of the conduction current, I = , where
Hence, I = V = 230 100 = 6.9 A = 6.9
Yes, the conduction current is equal to the displacement current.
Magnetic field is given as, B = , where
= Free space permeability = 4 N
= Maximum value of current =
r = distance between two plates on the axis = 3.0 cm = 0.03 m
then, B = 6.9 = 1.626 T
less
<p><strong>8.2 </strong>Radius of each circular plate, R = 6.0 cm = 0.06 m</p><p>Capacitance of parallel capacitor, C = 100 pF = 100 <span title="Click to copy mathml"><math><mo>×</mo><mi></mi><msup><mrow><mrow><mn>10</mn></mrow></mrow><mrow><mrow><mo>-</mo><mn>12</mn></mrow></mrow></msup></math></span> F</p><p>Supply voltage, V = 230 V</p><p>Angular frequency, <span title="Click to copy mathml"><math><mi>ω</mi></math></span> = 300 rad/s</p><p>rms value of the conduction current, I = <span title="Click to copy mathml"><math><mfrac><mrow><mrow><mi>V</mi></mrow></mrow><mrow><mrow><msub><mrow><mrow><mi>X</mi></mrow></mrow><mrow><mrow><mi>c</mi></mrow></mrow></msub></mrow></mrow></mfrac></math></span> , where <span contenteditable="false"> <math> <msub> <mrow> <mrow> <mi>X</mi> </mrow> </mrow> <mrow> <mrow> <mi>c</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal"> </mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">e</mi> <mo>=</mo> <mi mathvariant="normal"> </mi> <mfrac> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> <mrow> <mrow> <mi mathvariant="normal">ω</mi> <mi mathvariant="normal">C</mi> </mrow> </mrow> </mfrac> </math> </span></p><p>Hence, I = V <span title="Click to copy mathml"><math><mo>×</mo><mi>ω</mi><mo>×</mo><mi>C</mi></math></span> = 230 <span title="Click to copy mathml"><math><mo>×</mo><mn>300</mn><mo>×</mo></math></span> 100 <span title="Click to copy mathml"><math><mo>×</mo><mi></mi><msup><mrow><mrow><mn>10</mn></mrow></mrow><mrow><mrow><mo>-</mo><mn>12</mn></mrow></mrow></msup></math></span> = 6.9 <span title="Click to copy mathml"><math><mo>×</mo><msup><mrow><mrow><mn>10</mn></mrow></mrow><mrow><mrow><mo>-</mo><mn>6</mn></mrow></mrow></msup></math></span> A = 6.9<span contenteditable="false"> <math> <mi>μ</mi> <mi>A</mi> </math> </span></p><p>Yes, the conduction current is equal to the displacement current.</p><p>Magnetic field is given as, B = <span title="Click to copy mathml"><math><mfrac><mrow><mrow><msub><mrow><mrow><mi>μ</mi></mrow></mrow><mrow><mrow><mn>0</mn></mrow></mrow></msub><mi>r</mi></mrow></mrow><mrow><mrow><mn>2</mn><mi>π</mi><msup><mrow><mrow><mi>R</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup></mrow></mrow></mfrac><msub><mrow><mrow><mi>I</mi></mrow></mrow><mrow><mrow><mn>0</mn></mrow></mrow></msub></math></span> , where</p><p><span title="Click to copy mathml"><math><msub><mrow><mrow><mi>μ</mi></mrow></mrow><mrow><mrow><mn>0</mn></mrow></mrow></msub><mi></mi></math></span> = Free space permeability = 4 <span title="Click to copy mathml"><math><mi>π</mi><mo>×</mo><msup><mrow><mrow><mn>10</mn></mrow></mrow><mrow><mrow><mo>-</mo><mn>7</mn></mrow></mrow></msup></math></span> N <span title="Click to copy mathml"><math><msup><mrow><mrow><mi>A</mi></mrow></mrow><mrow><mrow><mo>-</mo><mn>2</mn></mrow></mrow></msup></math></span></p><p><span title="Click to copy mathml"><math><msub><mrow><mrow><mi>I</mi></mrow></mrow><mrow><mrow><mn>0</mn></mrow></mrow></msub></math></span> = Maximum value of current = <span title="Click to copy mathml"><math><mo>√</mo><mn>2</mn><mi>I</mi></math></span></p><p>r = distance between two plates on the axis = 3.0 cm = 0.03 m</p><p>then, B = <span title="Click to copy mathml"><math><mfrac><mrow><mrow><mn>4</mn><mi>π</mi><mo>×</mo><msup><mrow><mrow><mn>10</mn></mrow></mrow><mrow><mrow><mo>-</mo><mn>7</mn></mrow></mrow></msup><mo>×</mo><mn>0.03</mn></mrow></mrow><mrow><mrow><mn>2</mn><mi>π</mi><mo>×</mo><msup><mrow><mrow><mn>0.06</mn></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup></mrow></mrow></mfrac></math></span> <span title="Click to copy mathml"><math><mo>×</mo><mo>√</mo><mn>2</mn><mo>×</mo></math></span> 6.9 <span title="Click to copy mathml"><math><mo>×</mo><msup><mrow><mrow><mn>10</mn></mrow></mrow><mrow><mrow><mo>-</mo><mn>6</mn></mrow></mrow></msup></math></span> = 1.626 <span title="Click to copy mathml"><math><mo>×</mo><msup><mrow><mrow><mn>10</mn></mrow></mrow><mrow><mrow><mo>-</mo><mn>11</mn></mrow></mrow></msup></math></span> T</p>
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